The given partial differential equation is:

$pq + qx = y$

where $p = \frac{\partial u}{\partial x}$ and $q = \frac{\partial u}{\partial y}$, and we are looking for a solution for the function $u(x, y)$.

Step 1: Rewrite the equation

The given equation can be rewritten in terms of partial derivatives:

$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y} + \frac{\partial u}{\partial y} \cdot x = y$

which simplifies to:

$q p + qx = y$

This is a first-order linear partial differential equation. To solve it, we will first look for a method like the method of characteristics.

Step 2: Apply the method of characteristics

We can associate this with a system of ordinary differential equations (ODEs) for the characteristics. For the equation $pq + qx = y$, the characteristic equations are:

1. $\frac{dx}{ds} = q$
2. $\frac{dy}{ds} = x$
3. $\frac{du}{ds} = y$

From these, we can solve for the characteristic curves, and in turn, the general solution to the partial differential equation.

Would you like me to continue solving this in more detail or need clarification on the steps?